Why can't I persist binary(4) computed column?

I'm not sure what "http://www.sql-server-helper.com/" is, but this is from the current, official documentation:

The following built-in functions from other categories are always nondeterministic.
...
PARSENAME

Side note: it looks like PARSENAME was deterministic, at least on SQL Server 2005 (thanks for that link, jpa). So perhaps that other site just has outdated information on it.

You would need to re-implement this function using only deterministic functions (for instance, SUBSTRING and CHARINDEX).

Just to show that it can be deterministic (note that this implementation doesn't deal with general IP addresses, it's just an example):

CREATE OR ALTER FUNCTION dbo.fnBinaryIPv4(@ip AS VARCHAR(15)) RETURNS BINARY(4)
AS
BEGIN
    DECLARE @bin AS BINARY(4)

    SELECT @bin = CAST( CAST( PARSENAME( @ip, 4 ) AS INTEGER) AS BINARY(1))
                + CAST( CAST( PARSENAME( @ip, 3 ) AS INTEGER) AS BINARY(1))
                + CAST( CAST( PARSENAME( @ip, 2 ) AS INTEGER) AS BINARY(1))
                + CAST( CAST( PARSENAME( @ip, 1 ) AS INTEGER) AS BINARY(1))

    RETURN @bin
END
GO

SELECT OBJECTPROPERTY(OBJECT_ID('dbo.fnBinaryIPv4'), 'IsDeterministic') AS IsDeterministic;
GO

CREATE OR ALTER FUNCTION dbo.fnBinaryIPv4(@ip AS VARCHAR(15)) RETURNS BINARY(4)
WITH SCHEMABINDING
AS
BEGIN
    DECLARE @bin AS BINARY(4)

    SELECT @bin = CAST( CAST( SUBSTRING( @ip, 0, 3 ) AS INTEGER) AS BINARY(1))
                + CAST( CAST( SUBSTRING( @ip, 4, 3 ) AS INTEGER) AS BINARY(1))
                + CAST( CAST( SUBSTRING( @ip, 8, 3 ) AS INTEGER) AS BINARY(1))
                + CAST( CAST( SUBSTRING( @ip, 12, 3 ) AS INTEGER) AS BINARY(1))

    RETURN @bin
END
GO

SELECT OBJECTPROPERTY(OBJECT_ID('dbo.fnBinaryIPv4'), 'IsDeterministic') AS IsDeterministic;
GO